Because the x^4x4 coefficient is 11 we know the coefficient for the x^2x2 terms in the factor will also be 11:
(x^2 )(x^2 )(x2)(x2)
Because the constant is a negative and the coefficient for the xx term is a negative we know the sign for the constants in the factors will have one positive and one negative:
(x^2 + )(x^2 - )(x2+)(x2−)
Now we need to determine the factors which multiply to -4 and also add to -3:
1 xx -4 = -41×−4=−4; 1 - 4 = -31−4=−3 <- this IS the factor
(x^2 + 1)(x^2 - 4)(x2+1)(x2−4)
The factor (x^2 - 4)(x2−4) is a special form of the quadratic:
color(red)(x)^2 - color(blue)(y)^2 = (color(red)(x) + color(blue)(y))(color(red)(x) - color(blue)(y))x2−y2=(x+y)(x−y)
We can factor this term as:
(x^2 + 1)(color(red)(x) + color(blue)(2))(color(red)(x) - color(blue)(2))(x2+1)(x+2)(x−2)
